An automaton is synchronizing if there exists a word driving it into a definite state regardless of the starting state. For an n-state automaton the set of synchronizing words is a regular language that can be accepted by an automaton having $2^n-n$ states. Here, we study n-state automata over two input letters having the property that the minimal automata for their sets of synchronizing words have $2^n-n$ states, i.e., the minimal automata are maximal possible. We give a sufficient condition for this property that links it to completely reachable automata, non-trivial automaton congruences and the notion of uniform minimality. We apply our result to the family ${\mathcal{K}}_n$ of automata that was previously only conjectured to have this property.

如果存在一个词将自动机驱动到确定状态（无论起始状态如何），则自动机正在同步。对于n状态自动机，同步字集是一种常规语言，可以被具有以下特征的自动机接受：$2^n-n$状态。在这里，我们研究两个输入字母上的n状态自动机，其具有同步单词集的最小自动机具有的属性$2^n-n$状态，即最小自动机是最大可能的。我们给出了这个属性的充分条件，将其与完全可达自动机、非平凡自动机同余和一致极小性的概念联系起来。我们将我们的结果应用于家庭${\mathcal{K}}_n$以前仅推测具有此属性的自动机。